Akshay Shankar
2022-09-24
\[\small H = -t\sum_{\langle i, j \rangle} a_i^{\dagger}a_j\hspace{0.5cm}\longrightarrow\hspace{0.5cm}|\Psi_{SF}\rangle= \frac{1}{N!} (\sum_{i=1}^M a_i^{\dagger})^N |{0}\rangle \hspace{0.5cm}\]
Consider N bosons in M lattice sites.
10 bosons; 10 lattice sites.
\[\small \underbrace{H =
-t\sum_{\langle i, j \rangle} a_i^{\dagger}a_j + \frac{U}{2}\sum_i
n_i(n_i - 1)}_{\text{coupled lattice sites}}
\hspace{0.5cm}\longrightarrow\hspace{0.5cm} \underbrace{H \{\Psi \} =
\sum_i-zt \cdot (\Psi^*a_i + \Psi a_i^{\dagger} - |\Psi|^2) +
\frac{U}{2}n_i(n_i -1)}_{\text{de-coupled lattice sites}}\]
To find the equilibrium state of the system, we must minimize
\(G = H - \mu N + TS\).
\[\frac{\partial \langle G\rangle}{\partial \Psi} =
\frac{\partial \langle H - \mu N\rangle}{\partial \Psi}\Bigg|_{T=0K} =
0\]
\[ \Psi = \langle \psi_{gs} |\hat{a} |
\psi_{gs} \rangle\]
\[\text{Minimize G} \equiv \text{Self-consistently
diagonalize } H\{\Psi\}\]
\[\small \underbrace{H = -t\sum_{\langle i, j \rangle} a_i^{\dagger}a_j + \frac{U}{2}\sum_i n_i(n_i - 1)}_{\text{coupled lattice sites}} \hspace{0.5cm}\longrightarrow\hspace{0.5cm} \underbrace{H \{\Psi_i \} = \sum_C H_{exact} + \sum_{C, C'}H_{MFT}\{ \Psi_i \}}_{\text{de-coupled clusters of sites}}\]
\[\small H = -t\sum_{\langle i, j \rangle}
a_i^{\dagger}a_j + \frac{U}{2}\sum_i n_i(n_i - 1) + V\sum_{\langle i, j
\rangle} n_i n_j\] \[\hat{n}_i =
\rho_i + \delta \hat{n}_i\hspace{0.5cm}\Bigg\downarrow
\hspace{0.5cm}\hat{a}_i = \Psi_i + \delta\hat{a}_i\] \[\small H_A \{\Psi_A, \Psi_B,\rho_A, \rho_B \} =
-zt \cdot (\Psi_B^*a_A + \Psi_B a_A^{\dagger} - \Psi_A^*\Psi_B) +
zV\cdot(\rho_Bn_A - \rho_A\rho_B) + \frac{U}{2}n_A(n_A -1) \\
\small H_B \{\Psi_A, \Psi_B,\rho_A, \rho_B \} = -zt \cdot (\Psi_A^*a_B +
\Psi_A a_B^{\dagger} - \Psi_B^*\Psi_A) + zV\cdot(\rho_An_B -
\rho_B\rho_A) + \frac{U}{2}n_B(n_B -1)\]
\[\small H \{\Psi_A, \Psi_B,\rho_A, \rho_B\}=
\sum_{i \in A} H_i + \sum_{j \in B} H_j\]
Solve self-consistently: \(\hspace{0.5cm}\Psi_i = \langle \psi_{gs,
i}|\hat{a}_i |\psi_{gs, i}\rangle \hspace{0.5cm};\hspace{0.5cm} \rho_i =
\langle \psi_{gs, i}|\hat{n}_i |\psi_{gs, i}\rangle \hspace{0.5cm};
\hspace{0.5cm} i \in \{A, B\}\)
Quantum Monte Carlo methods
\[\underbrace{\hat{H}_0 = \frac{U}{2} \sum_i n_i(n_i-1) - \mu\sum_in _i}_{\text{diagonal in } \{|n\rangle\}} \hspace{1cm} \text{and} \hspace{1cm} \underbrace{\hat{V} = t\sum_{\langle i, j \rangle} a_i^{\dagger}a_j}_{\text{off-diagonal in } \{|n\rangle\}}\]
\[Z \equiv \sum_{m=0}^{\infty} \sum_{i_1,..,i_m} e^{-\beta \epsilon_1} \int_0^{\beta} d\tau_1 \dots d\tau_m\int_0^{\tau_{m-1}} (e^{-\tau_1\epsilon_1}\langle i_1|V|i_2\rangle e^{\tau_1\epsilon_2})\dots(e^{-\tau_m\epsilon_m}\langle i_m | V|i_1\rangle e^{\tau_m\epsilon_1})\]
Evaluate the path integral to arbitrary accuracy.
Train the network weights to minimize \(\langle \hat{H} \rangle\).